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The Sperner Game

Text III

We suggest that you go back to the playing board, making sure you are not playing against the computer, and click on the path button.

On the small triangle in the upper left-hand corner, you will find a counter-clockwise orientation, as shown by the arrows, and a colored arrow going from the red disk to the green one.

Returning to the large triangle, whenever there is a change from a red to a green disk, along the right-hand side of the triangle (with respect to the given orientation), a cross arrow is drawn, with the red disk on its left-hand side and the green disk on its right-hand side.

Notice that, in this example, in between the two disks at the center of the right-hand edge of the triangle, no arrow was drawn, since the change in color from green to red does not respect the orientation we are considering but rather the opposite orientation.

If the path thus started can be continued, in such a way that still a red disk appears to the left and a green one to the right, then a new arrow will be drawn (place the mouse pointer over the image):

What do you thing happens to the selected path in the image above if, over the yet uncolored disk to the left of the green one, a red, a green or a blue disk is placed, respectively? When does the path continue, and where does it go through?

Notice that the last configuration (blue) corresponds to a losing position, as a small triangle with three colors has appeared. It is only in this case of the blue disk that the path cannot proceed: it ends inside the small tricolored triangle. After you have drawn your conclusions, you can check them by successively placing the pointer of the mouse over each of the three images.

Could it be that every path can only come to an end inside of such a tricolored triangle?

The following example shows that the answer to this question is negative:

In this example, two of the paths find their way out of the right-hand side of the large triangle - the side where all paths had started. In both of these two cases, at the place where the path emerges outside of the triangle, we have, as usual, a red disk at the left of the path and a green disk at its right. But now, at the end of the path, the change from red to green has an orientation, with respect to the side of the large triangle, which is opposite to the orientation we were considering at the beginning of the path.

Let us consider a path that does not exit the large triangle through its right-hand side. What could possibly happen to it? Could it go twice through the same small triangle, i.e., is it possible to have a path like the one represented bellow?

No such path is possible, and the reason for that is simple:

look at the first small triangle where the path supposedly goes through twice (in the image, this is the triangle below the question mark);

considering the first time the path goes through the triangle (horizontally, from right to left), the colors have to be as follows:

Hence, the path cannot go into that triangle again through the other (top) side, as both disks are green.

Summing up:

if there is a path that goes into the right-hand side of the large triangle and does not exit this triangle through that same side, then:

it cannot exit through any of the other two sides of the large triangle, as neither side contains both colors (red and green);

it does not go twice through the same small triangle;

as there are finitely many small triangles, the path must come to an end.

But we have seen before that the only case when a path could not leave the inside of a small triangle is when that triangle has all three colors.

What can we thus conclude from the above?

We can conclude that, in order to prove that there is always a small tricolored triangle, we need only show that there has to be a path going into the right-hand side of the large triangle but not going out from that same side.

Why don't you return to the game and, for different values of n, color the right-hand side of the large triangle in a variety of ways. For each choice of colored disks, notice the changes in color that occur, count how many of these changes correspond to paths that start on that side and how many could be path endings. What do you conclude by comparing these numbers? Do you or do you not conclude that, in the examples that you considered, there is at least one path that starts on that side but does not terminate on that side?

If you wish, do that same comparison between the number of paths going in and the number of (possible) paths going out of the right-hand side of the triangle, for the example shown below (and for those others that you obtain by placing the mouse pointer over the image).

After you have made these observations (and not sooner!), access the final page on this subject.